Abelian variety isogeny class 2.89.aw_ji over $\F_{89}$

• Label: 2.89.aw_ji
• Base field: $\F_{89}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$1 - 22 x + 242 x^{2} - 1958 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.0585349918242$, $\pm0.441465008176$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{57})\)
Galois group:	$C_2^2$
Jacobians:	$64$
Isomorphism classes:	126

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6184$	$62730496$	$496593708712$	$3935115128406016$	$31180550303642832424$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$7922$	$704420$	$62718750$	$5583849988$	$496981290962$	$44231343944548$	$3936588780734014$	$350356402627473860$	$31181719929966183602$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 64 curves (of which all are hyperelliptic):

• $y^2=53 x^6+24 x^5+24 x^4+47 x^3+28 x^2+32 x+60$
• $y^2=62 x^6+27 x^5+77 x^4+58 x^3+72 x^2+68 x+82$
• $y^2=59 x^6+21 x^5+59 x^4+50 x^3+52 x^2+57 x+25$
• $y^2=60 x^6+x^5+39 x^4+22 x^3+51 x^2+59 x+77$
• $y^2=23 x^6+65 x^5+85 x^4+28 x^3+68 x^2+68 x+15$
• $y^2=8 x^6+75 x^5+56 x^4+33 x^3+81 x^2+42 x+27$
• $y^2=85 x^6+43 x^5+68 x^4+35 x^3+75 x+6$
• $y^2=33 x^6+45 x^5+17 x^4+15 x^3+69 x^2+27 x+19$
• $y^2=6 x^6+48 x^5+63 x^4+49 x^3+25 x^2+86 x+28$
• $y^2=75 x^6+42 x^5+66 x^4+32 x^3+72 x^2+75 x+54$
• $y^2=39 x^6+85 x^5+10 x^4+23 x^3+80 x^2+41 x+75$
• $y^2=23 x^6+72 x^5+54 x^4+8 x^3+30 x^2+42 x+29$
• $y^2=19 x^6+60 x^5+56 x^4+12 x^3+42 x^2+44 x+6$
• $y^2=33 x^6+63 x^5+2 x^4+x^3+69 x^2+54 x+65$
• $y^2=55 x^6+82 x^5+88 x^4+49 x^3+41 x^2+18 x+43$
• $y^2=53 x^6+88 x^5+22 x^4+51 x^3+56 x^2+40 x+74$
• $y^2=52 x^6+68 x^5+40 x^4+27 x^3+35 x^2+32 x+27$
• $y^2=2 x^6+86 x^5+15 x^4+38 x^3+52 x^2+87 x+76$
• $y^2=69 x^6+66 x^5+67 x^4+79 x^3+69 x^2+40 x+34$
• $y^2=54 x^6+58 x^5+25 x^4+38 x^3+70 x^2+69 x+77$
• and

• $y^2=56 x^6+16 x^5+6 x^4+22 x^3+88 x^2+54 x+26$
• $y^2=2 x^6+14 x^5+15 x^4+35 x^3+88 x^2+40 x+12$
• $y^2=19 x^6+53 x^5+69 x^3+64 x^2+27 x+41$
• $y^2=24 x^6+14 x^5+31 x^4+59 x^3+47 x^2+67 x+25$
• $y^2=3 x^6+76 x^5+20 x^4+18 x^3+16 x^2+40 x+67$
• $y^2=x^6+67 x^5+63 x^4+70 x^3+8 x^2+61 x+63$
• $y^2=51 x^6+48 x^5+6 x^4+88 x^3+85 x^2+56 x+12$
• $y^2=68 x^6+56 x^5+83 x^4+36 x^3+75 x^2+68 x+23$
• $y^2=50 x^6+55 x^5+45 x^4+57 x^3+52 x^2+75 x+28$
• $y^2=68 x^6+54 x^5+88 x^4+6 x^3+47 x^2+14 x+15$
• $y^2=77 x^6+54 x^5+70 x^4+19 x^3+44 x^2+32 x+86$
• $y^2=63 x^6+32 x^5+66 x^4+70 x^3+84 x^2+26 x+56$
• $y^2=48 x^6+21 x^5+35 x^4+35 x^2+68 x+48$
• $y^2=65 x^6+7 x^5+27 x^4+35 x^3+x^2+25 x+39$
• $y^2=14 x^6+39 x^5+81 x^4+35 x^3+3 x^2+59 x+32$
• $y^2=85 x^6+7 x^5+39 x^4+49 x^3+50 x^2+15 x+84$
• $y^2=27 x^6+84 x^5+76 x^4+15 x^3+27 x^2+38 x+81$
• $y^2=32 x^6+69 x^5+29 x^4+43 x^3+36 x^2+24 x+69$
• $y^2=6 x^6+9 x^5+40 x^4+57 x^3+64 x^2+54 x+20$
• $y^2=59 x^6+14 x^5+20 x^4+30 x^3+73 x^2+20 x+79$
• $y^2=29 x^6+75 x^5+76 x^4+76 x^3+65 x^2+78 x+26$
• $y^2=3 x^6+13 x^5+35 x^4+29 x^3+28 x^2+35 x+32$
• $y^2=42 x^6+6 x^5+78 x^4+72 x^3+75 x^2+61 x+39$
• $y^2=52 x^6+42 x^5+70 x^4+26 x^3+53 x^2+41 x+13$
• $y^2=71 x^5+6 x^4+59 x^3+19 x^2+27 x+39$
• $y^2=78 x^6+63 x^5+65 x^4+88 x^3+48 x^2+40 x+24$
• $y^2=35 x^6+87 x^5+x^4+3 x^3+34 x^2+14 x+46$
• $y^2=88 x^5+57 x^4+33 x^3+84 x^2+42 x+19$
• $y^2=46 x^6+52 x^5+61 x^4+28 x^3+13 x^2+25 x+20$
• $y^2=71 x^6+18 x^5+36 x^4+82 x^3+57 x^2+20 x+48$
• $y^2=57 x^6+82 x^5+43 x^4+77 x^3+32 x^2+x+51$
• $y^2=64 x^6+3 x^5+29 x^4+12 x^3+67 x^2+17 x+31$
• $y^2=3 x^6+65 x^5+33 x^4+13 x^3+5 x^2+68 x+19$
• $y^2=7 x^6+34 x^5+25 x^4+52 x^3+78 x^2+45 x+75$
• $y^2=9 x^6+41 x^5+48 x^4+19 x^3+77 x^2+56 x+6$
• $y^2=71 x^6+50 x^5+28 x^4+66 x^3+87 x^2+53 x+86$
• $y^2=86 x^6+67 x^5+15 x^4+74 x^3+72 x^2+16 x+8$
• $y^2=44 x^6+x^5+47 x^4+47 x^3+67 x^2+11 x+64$
• $y^2=51 x^6+4 x^5+46 x^4+50 x^3+72 x^2+42 x+16$
• $y^2=61 x^6+26 x^5+67 x^4+59 x^3+52 x^2+80 x+88$
• $y^2=17 x^6+34 x^5+54 x^4+42 x^3+10 x^2+49 x+3$
• $y^2=85 x^6+49 x^5+28 x^4+61 x^3+45 x^2+54 x+82$
• $y^2=26 x^6+15 x^5+88 x^4+18 x^3+9 x^2+64 x+69$
• $y^2=76 x^6+62 x^5+27 x+76$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89^{4}}$.

Endomorphism algebra over $\F_{89}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{57})\).

Endomorphism algebra over $\overline{\F}_{89}$

The base change of $A$ to $\F_{89^{4}}$ is 1.62742241.arju^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-57}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{89^{2}}$

  The base change of $A$ to $\F_{89^{2}}$ is the simple isogeny class 2.7921.a_arju and its endomorphism algebra is \(\Q(i, \sqrt{57})\).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.89.w_ji	$2$	(not in LMFDB)
2.89.w_ji	$4$	(not in LMFDB)
2.89.a_acm	$8$	(not in LMFDB)